Extreme Financial Risks (2006)

Extreme Financial Risks
From Dependence to Risk Management
Malevergne, Yannick, Sornette, Didier
2006, XVI, 312 p. 62 illus., Softcover
ISBN: 3-540-27264-X
Springer, Heidelberg

Preface: idiosyncratic and collective extreme risks

Modern western societies have a paradoxical relationship with risks. On the one hand, there is the utopian quest for a zero risk society. On the other hand, human activities may increase risks of all kinds, from collaterals of new technologies to global impacts on the planet. The characteristic multiplication of major risks in modern society and its reflexive impact on its development is at the core of the concept of the "Risk Society''. Correlatively, our perception of risk has evolved so that catastrophic events (earthquakes, floods, droughts, storms, hurricanes, volcanic eruptions, and so on) are no more systematically perceived as unfair outcomes of an implacable destiny. Catastrophes may also result from our own technological developments whose complexity may engender major industrial disasters such as Bhopal, Tchernobyl, AZT ... as well as irreversible global changes such as global warming leading to climatic disruptions or epidemics from new bacterial and viral mutations. The proliferation of new sources of risks imposes new responsibilities concerning their determination, understanding and management. Government organizations as well as private institutions such as industrial companies, insurance companies and banks which have to face such risks, in their role of regulators or of risk bearers, must ensure that the consequences of extreme risks are supportable without endangering the institutions in charge of bearing these risks.

In the financial sector, crashes probably represent the most striking events among all possible extreme phenomena, with an impact and frequency that has been increasing in the last two decades. Consider the worldwide crash in October 1987 which evaporated more than one thousand billion dollars in a few days or the more recent collapse of the internet bubble bubble} in which more than one-third of the world capitalization of 1999 disappeared after March 2000. Finance and stock markets are based on the fluid convertibility of stocks into money and vice-versa. Thus, to work well, money is requested to be a reliable standard of value, that is, an effective store of value, hence the concerns with the negative impacts of inflation. Similarly, investors look at the various financial assets as carriers of value, like money, but with additional return potentials (accompanied with downturn risks). But for this view to hold so as to promote economic development, fluctuations in values need to be tamed to minimize the risk of losing a lifetime of savings, or to avoid the risks of losing the investment potential of companies or even to prevent economic and social recessions in whole countries (consider the situation of California after 2002 with a budget gap representing more than one-fourth of the entire State budget resulting essentially from the losses of financial and tax incomes following the collapse of the Internet bubble). It is thus highly desirable to have the tools for monitoring, understanding and limiting the extreme risks of financial markets. Fully aware of these problems, the worldwide banking organizations have promoted a series of advices and norms, known as the recommendations of the Basle committee. The Basle committee has proposed models for the internal management of risks and the imposition of minimum margin requirements commensurate with the risk exposures. However, some criticisms have found these recommendations to be ill-adapted or even destabilizing. This controversy underlines the importance of a better understanding of extreme risks, of their consequences and of ways to prevent or at least minimize them.

In our opinion, tackling this challenging problem requires to decompose it in two main parts. First, it is essential to be able to accurately quantify extreme risks. This calls for the development of novel statistical tools going significantly beyond the Gaussian paradigm which underpins the standard framework of classical financial theory inherited from Bachelier, Markowitz and Black and Scholes among others. Second, the existence of extreme risks must be considered in the context of the practice of risk management itself, which leads to ask whether extreme risks can be diversified away similarly to standard risks according to the mean-variance approach. If the answer to this question is negative as can be surmised for numerous concrete empirical evidences, it is necessary to develop new concepts and tools for the construction of portfolios with minimum (but unavoidable) exposition of extreme risks. One can think of mixing equities and derivatives, as long as derivatives themselves do not add an extreme risk component and can really provide an insurance against extreme moves, which has been far from true in recent dramatic instances such as the crash of October 1987. Another approach could involve mutualism as in insurance.

Risk management, and to the same extent portfolio management, thus requires a precise and rigorous analysis of the distribution of the returns of
the portfolio of risks. Taking into account the moderate sizes of standard portfolios (from tens to thousands of assets typically) and the non-Gaussian nature of the distributions of the returns of assets constituting the portfolios, the distributions of the returns of typical portfolios are far from Gaussian, in contradiction with the expectation from a naive use of the central limit theorem. This breakdown of universality then requires a careful estimation of the specific case-dependent distribution of the returns of a given portfolio. This can be done directly using the time series of the returns of the portfolio for a given capital allocation. A more constructive approach consists in estimating the joint distribution of the returns of all assets constituting the portfolio. The first approach is much simpler and rapid to implement since its requires solely the estimation of a monovariate distribution. However, it lacks generality and power by neglecting the observable information available from the basket of all returns of the assets. Only the multivariate distribution of the returns of the assets embodies the general information of all risk components and their dependence across assets. However, the two approaches become equivalent in the following sense: the knowledge of the distribution of the returns for all possible portfolios for all possible allocations of capital between assets is equivalent to the knowledge of the multivariate distributions of the asset returns. {\it In fine}, the second approach appears preferable on a general basis and is the method mobilizing the largest efforts both in academia and in the private sector.

However, the frontal attack aiming at the determination of the multivariate distribution of the asset returns is a challenging task and, in our opinion, much less instructive and useful than the separate studies of the marginal distributions of the asset returns on the one hand and the dependence structure of these assets on the other hand. In this book, we emphasize this second approach, with the objective of characterizing as faithfully as possible the diverse origins of risks: the risks stemming from each individual asset and the risks having a collective origin. This requires to determine (1) the distributions of returns at different time scales or more generally the stochastic process underlying the assets prices dynamics and (2) the nature and properties of dependences between the different assets.

The present book offers an original and thorough treatment of these two domains, focusing mainly on the concepts and tools that remain valid for large and extreme price moves. Its originality lies in detailed and thorough presentations of the state of the art on (i) the different distributions of financial returns for various applications (VaR, stress testing) and (ii) the most important and useful measures of dependences, both unconditional and conditional and a study of the impact of conditioning on the size of large moves on the measure of extreme dependences. A large emphasis is thus put on the theory of copulas, their empirical testing and calibration, as they offer intrinsic and complete measures of dependences. Many of the results presented here are novel and have not been published or have been recently obtained by the authors or their colleagues. We would like to acknowledge, in particular, the fruitful and inspiring discussions and collaborations with J.V. Andersen, U. Frisch, J.-P. Laurent, J.-F. Muzy and V.F. Pisarenko.

Chapters

Chap. 1 describes a general framework to develop "coherent measures'' of risks. It also addresses the origins of risks and of dependence between assets in financial markets, from the CAPM (capital asset pricing model) generalized to the non-Gaussian case with heterogeneous agents, the APT (arbitrage pricing theory), the factor models to the complex system view suggesting an emergent nature for the risk-return trade-off.

Chap. 2 addresses the problem of the precise estimation of the probability of extreme events, based on a description of the distribution of asset returns endowed with heavy tails. The challenge is thus to characterize precisely these heavy tails, which are characterized by poor sampling (large events are rare). A major difficulty is to neither underestimate (Gaussian error) or overestimate (heavy tail hubris) the extreme events. The quest for a precise quantification opens the door to model errors, which can be partially circumvented by using several families of distributions whose detailed comparisons allow one to discern the sources of uncertainty and errors. Chap. 2 thus discusses several classes of heavy tailed distributions: regularly varying distributions (i.e., with asymptotic power law tails), stretched-exponential distributions (also known as Weibull or sub-exponentials) as well as log-Weibull distributions which extrapolate smoothly between these different families.

The second element of the construction of multivariate distributions of asset returns, addressed in Chaps. 3-6, is to quantify the dependence structure of the asset returns. Indeed, large risks are not due solely to the heavy tails of the distribution of returns of individual assets but may result from a collective behavior. This collective behavior can be completely described by mathematical objects called copulas, introduced in Chap. 3, which fully embody the dependence between asset returns.

Chap. 4 describes synthetic measures of dependences, contrasting and linking them with the concept of copulas. It also presents an original estimation method of the coefficient of tail dependence, defined roughly speaking as the probability for an asset to lose a large amount knowing that another asset or the market has also dropped significantly. This tail dependence is of great interest because it addressed in a straightforward way the fundamental question whether extreme risks can be diversified away or not by aggregation in portfolios. Either the tail dependence coefficient is zero and the extreme losses occur asymptotically independently, which opens the possibility of diversifying them away. Or the tail dependence coefficient is non-zero and extreme losses are fundamentally dependent and it is impossible to completely remove extreme risks. The only remaining strategy is to develop portfolios that minimize the collective extreme risks, thus generalizing the mean-variance to a mean-extreme theory.

Chap. 5 presents the main methods for estimating copulas of financial assets. It shows that the empirical determination of a copula is quite delicate with significant risks of model errors, especially for extreme events. Specific studies of the extreme dependence are thus required.

Chap. 6 presents a general and thorough discussion of different measures of conditional dependences (where the condition can be on the size(s) of one or both returns for two assets). Chap. 6 thus sheds new lights on the variations of the strength of dependence between assets as a function of the sizes of the analyzed events. As a startling concrete application of conditional dependences, the phenomenon of contagion during financial crises is discussed in details.

Chap. 7 presents a synthesis of the six previous chapters and then offers suggestions for future work on dependence and risk analysis, including time-varying measures of extreme events, endogeneity versus exogeneity, regime switching, time-varying lagged dependence and so on.

Audience

This book has been written with the ambition to be useful to (a) the student looking for a general and in-depth introduction to the field, (b) financial engineers, economists, econometricians, actuarial professionals and researchers and mathematicians looking for a synoptic view comparing the pros and cons of different modeling strategies and (c) quantitative practitioners for the insights offered on the subtleties and many dimensional components of both risk and dependence. The content of this book will also be useful to the broader scientific community in the Natural Sciences, interested in quantifying the complexity of many physical, geophysical, biophysical, etc., processes, with a mounting emphasis on the role and importance of extreme phenomena and their non-standard dependences.